The Hadamard Product of Moment Sequences

We define the (Hadamard) product $s\odot t := (s_\alpha\cdot t_\alpha)_{\alpha\in\mathbb{N}_0^n}$ of two sequences $s = (s_\alpha)_{\alpha\in\mathbb{N}_0^n}$ and $t = (t_\alpha)_{\alpha\in\mathbb{N}_0^n}$. If $s$ is represented by a measure $\mu$ and $t$ by a measure $\nu$, i.e., both are moment sequences, then we give an explicit measure $\mu\odot\nu$ that represents $s\odot t$, i.e., $s\odot t$ is a moment sequence. We pick a result from Moment-sequence transforms (J. Eur. Math. Soc. 24 (2022), 3109--3160) and show how the multiplication $s\odot t$ of moment sequences simplifies the proof and also gives deeper insight by providing the representing measure $\mu\odot\nu$.